Quantum Computing: An Applied Approach

Description

This book integrates the foundations of quantum computing with a hands-on coding approach to this emerging field; it is the first to bring these elements together in an updated manner. This work is suitable for both academic coursework and corporate technical training. The second edition includes extensive updates and revisions, both to textual content and to the code. Sections have been added on quantum machine learning, quantum error correction, Dirac notation and more. This new edition benefits from the input of the many faculty, students, corporate engineering teams, and independent readers who have used the first edition. This volume comprises three books under one cover: Part I outlines the necessary foundations of quantum computing and quantum circuits. Part II walks through the canon of quantum computing algorithms and provides code on a range of quantum computing methods in current use. Part III covers the mathematical toolkit required to master quantum computing. Additional resources include a table of operators and circuit elements and a companion GitHub site providing code and updates. Jack D. Hidary is a research scientist in quantum computing and in AI at Alphabet X, formerly Google X. Contents Preface to the Second Edition Preface to the First Edition Acknowledgements Navigating this Book Part I Foundations CHAPTER 1 Superposition, Entanglement and Reversibility 1.1 Superposition and Entanglement 1.2 The Born Rule 1.3 Schrödinger’s Equation 1.4 The Physics of Computation CHAPTER 2 A Brief History of Quantum Computing 2.1 Early Developments and Algorithms 2.2 Shor and Grover 2.3 Defining a Quantum Computer CHAPTER 3 Qubits, Operators and Measurement Quantum Circuit Diagrams 3.1 Quantum Operators Unary Operators Binary Operators Ternary Operators 3.2 Comparison with Classical Gates 3.3 Universality of Quantum Operators 3.4 Gottesman-Knill and Solovay-Kitaev 3.5 The Bloch Sphere 3.6 The Measurement Postulate 3.7 Computation-in-Place CHAPTER 4 Complexity Theory 4.1 Problems vs. Algorithms 4.2 Time Complexity 4.3 Complexity Classes 4.4 Quantum Computing and the Church-Turing Thesis Part II Hardware and Applications CHAPTER 5 Building a Quantum Computer 5.1 Assessing a Quantum Computer 5.2 Neutral Atoms 5.3 NMR 5.4 NV Center-in-Diamond 5.5 Photonics Semiconductor quantum transistor Topological photonic chip 5.6 Spin Qubits 5.7 Superconducting Qubits 5.8 Topological Quantum Computation 5.9 Trapped Ion 5.10 Summary CHAPTER 6 Development Libraries for Quantum Computer Programming 6.1 Quantum Computers and QC Simulators 6.2 Cirq 6.3 Qiskit 6.4 Forest 6.5 Quantum Development Kit 6.6 Dev Libraries Summary Using the Libraries Other Development Libraries 6.7 Additional Quantum Programs Bell States Gates with Parameters CHAPTER 7 Teleportation, Superdense Coding and Bell’s Inequality 7.1 Quantum Teleportation 7.2 Superdense Coding 7.3 Code for Quantum Teleportation and Superdense Communication 7.4 Bell Inequality Test Summary CHAPTER 8 The Canon: Code Walkthroughs 8.1 The Deutsch-Jozsa Algorithm 8.2 The Bernstein-Vazirani Algorithm The Bernstein-Vazirani Algorithm 8.3 Simon’s Problem 8.4 Quantum Fourier Transform 8.5 Shor’s Algorithm RSA Cryptography The Period of a Function Period of a Function as an Input to a Factorization Algorithm Classical order finding Quantum order finding Quantum arithmetic operations in Cirq Modular exponential arithmetic operation Using the modular exponential operation in a circuit Classical post-processing Quantum order finder The complete factoring algorithm 8.6 Grover’s Search Algorithm Grover’s Algorithm in Qiskit 3-Qubit Grover’s Algo Summary CHAPTER 9 Quantum Computing Methods 9.1 Variational Quantum Eigensolver VQE with Noise More Sophisticated Ansatzes 9.2 Quantum Chemistry 9.3 Quantum Approximate Optimization Algorithm (QAOA) Example Implementation of QAOA 9.4 Machine Learning on Quantum Processors 9.5 Quantum Phase Estimation Implemention of QPE 9.6 Solving Linear Systems Description of the HHL Algorithm Example Implementation of the HHL Algorithm 9.7 Quantum Random Number Generator 9.8 Quantum Walks Implementation of a QuantumWalk 9.9 Unification Framework for Quantum Algorithms (QSVT) 9.10 Dequantization 9.11 Summary CHAPTER 10 Applications and Quantum Supremacy 10.1 Applications Quantum Simulation and Chemistry Sampling from Probability Distributions Linear Algebra Speedup with Quantum Computers Optimization Tensor Networks 10.2 Quantum Supremacy Random Circuit Sampling Other Problems for Demonstrating Quantum Supremacy Quantum Advantage and Beyond Classical Computation 10.3 Quantum Error Correction Context and Importance Important Preliminaries Motivating Example: The Repetition Code The Stabilizer Formalism 10.4 Doing Physics with Quantum Computers Conclusion Part III Toolkit CHAPTER 11 Mathematical Tools for Quantum Computing I 11.1 Introduction and Self-Test 11.2 Linear Algebra Vectors Introduction to Dirac Notation Basic Vector Operations The Norm of a Vector The Dot Product 11.3 The Complex Numbers and the Inner Product Complex Numbers The Inner Product as a Refinement of the Dot Product The Polar Coordinate Representation of a Complex Number 11.4 A First Look at Matrices Basic Matrix Operations The Identity Matrix Transpose, Conjugate and Trace Matrix Exponentiation 11.5 The Outer Product and the Tensor Product The Outer Product as a Way of Building Matrices The Tensor Product 11.6 Set Theory The Basics of Set Theory The Cartesian Product Relations and Functions Important Properties of Functions 11.7 The Definition of a Linear Transformation 11.8 How to Build a Vector Space From Scratch Groups Rings Fields The Definition of a Vector Space Subspaces 11.9 Span, Linear Independence, Bases and Dimension Span Linear Independence Bases and Dimension Orthonormal Bases CHAPTER 12 Mathematical Tools for Quantum Computing II 12.1 Linear Transformations as Matrices 12.2 Matrices as Operators An Introduction to the Determinant The Geometry of the Determinant Matrix Inversion 12.3 Eigenvectors and Eigenvalues Change of Basis 12.4 Further Investigation of Inner Products The Kronecker Delta Function as an Inner Product 12.5 Hermitian Operators Why We Can’t Measure with Complex Numbers Hermitian Operators Have Real Eigenvalues 12.6 Unitary operators 12.7 The Direct Sum and the Tensor Product The Direct Sum The Tensor Product 12.8 Hilbert Space Metrics, Cauchy Sequences and Completeness An Axiomatic Definition of the Inner Product The Definition of Hilbert Space 12.9 The Qubit as a Hilbert Space CHAPTER 13 Mathematical Tools for Quantum Computing III 13.1 Boolean Functions 13.2 Logarithms and Exponentials 13.3 Euler’s Formula CHAPTER 14 Dirac Notation 14.1 Vectors 14.2 Vector operations Inner and Outer Products 14.3 Tensor Products 14.4 Notation for PDF and Expectation Value CHAPTER 15 Table of Quantum Operators and Core Circuits Works Cited Index